Fundamentals of Electric Circuits - University of Anbar · 2020. 3. 14. · Fundamentals of EE II...

Fundamentals of EE II

EE1302 Dr. Naser Al-Falahy

University of Anbar College of Engineering Dept. of Electrical Engineering

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University of Anbar

College of Engineering

Department of Electrical Engineering

Fundamentals of

Electric Circuits

Second Course

Dr. Naser Al-Falahy




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WEEK1

Chapter 6

Capacitors and inductors

6.1Introduction So far we have limited our study to resistive circuits. In this chapter, we shall introduce two new and important passive linear circuit elements: the capacitor and the inductor. Unlike resistors, which dissipate energy, capacitors and inductors do not dissipate but store energy, which can be retrieved at a later time. For this reason, capacitors and inductors are called storage elements 6.2Capacitors A capacitor is a passive element designed to store energy in its electric field. Besides resistors, capacitors are the most common electrical Components. Capacitors are used extensively in electronics, communications, computers, and power systems. For example, they are used in the tuning circuits of radio receivers and as dynamic memory elements in computer systems.

In many practical applications, the plates may be aluminum foil while the dielectric may be air, ceramic, paper, or mica. When a voltage source is connected to the capacitor, as in Fig. 6.2, the source deposits a positive charge q on one plate and a negative charge on the other. The capacitor is




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said to store the electric charge. The amount of charge stored, represented by q, is directly proportional to the applied voltage so that

whereC, the constant of proportionality, is known as the capacitance of the capacitor. The unit of capacitance is the farad (F),

Although the capacitance C of a capacitor is the ratio of the charge q per plate to the applied voltage it does not depend on q or v It depends on the physical dimensions of the capacitor. For example, for the parallel-plate capacitor shown in Fig. 6.1, the capacitance is given by

where A is the surface area of each plate, d is the distance between the plates, and is Ԑ the permittivity of the dielectric material between the plates. Although Eq. (6.2) applies to only parallel-plate capacitors, we may infer from it that, in general, three factors determine the value of the capacitance: 1. The surface area of the plates—the larger the area, the greater the capacitance. 2. The spacing between the plates—the smaller the spacing, the greater the capacitance. 3. The permittivity of the material—the higher the permittivity, the greater the capacitance. Capacitors are commercially available in different values and types. Typically, capacitors have values in the Pico farad (pF) to microfarad (μF)range. They are described by the dielectric material they are made of and by whether they are of fixed or variable type. Figure 6.3 shows the circuit symbols for fixed and variable capacitors. Note that according to the passive sign convention, if v >0 and i >0 or v < 0 if and i< 0 the capacitor is being charged, and if v . i < 0,the capacitor is discharging.




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capacitors are used to block dc, pass ac, shift phase, store energy, start motors, and suppress noise . Variable capacitors are used in radio receivers allowing one to tune to various stations. To obtain the current-voltage relationship of the capacitor, we take the derivative of both sides of Eq. (6.1). Since




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Equation (6.9) or (6.10) represents the energy stored in the electric field that exists between the plates of the capacitor. This energy can be retrieved, since an ideal capacitor cannot dissipate energy. In fact, the word capacitor is derived from this element’s capacity to store energy in an electric field. We should note the following important properties of a capacitor: 1. Note from Eq. (6.4) that when the voltage across a capacitor is not changing with time (i.e., dc voltage), the current through the capacitor is zero. Thus

However, if a battery (dc voltage) is connected across a capacitor, the capacitor charges. 2. The voltage on the capacitor must be continuous

The capacitor resists an abrupt change in the voltage across it. According to Eq. (6.4), a discontinuous change in voltage requires an infinite current, which is physically impossible. For example, the voltage across a capacitor may take the form shown in Fig. 6.7(a), whereas it is not physically possible for the capacitor voltage to take the form shown in Fig. 6.7(b) because of the abrupt changes. Conversely, the current through a capacitor can change instantaneously




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3. The ideal capacitor does not dissipate energy. It takes power from the circuit when storing energy in its field and returns previously stored energy when delivering power to the circuit

4. A real, nonideal capacitor has a parallel-model leakage resistance, as shown in Fig. 6.8. The leakage resistance may be as high as 100 MΩ and can be neglected for most practical applications. For this reason, we will assume ideal capacitors .

Example (a) Calculate the charge stored on a 3-pF capacitor with 20 V across it. (b) Find the energy stored in the capacitor.




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Practice Problem What is the voltage across a 4.5 μF capacitor if the charge on one plate is 0.12 mC? How much energy is stored

Answer: 26.67 A, 1.6 mJ. Example The voltage across a 5 μF capacitor is v(t) = 10 cos 6000t V Calculate the current through it.

Practice Problem




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Example

Practice Problem

Example




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Determine the current through a 200 μF capacitor whose voltage is shown in Fig.




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Practice Problem

An initially uncharged 1-mF capacitor has the current shown in Fig. across it. Calculate the voltage across it at t=2ms and t=5 ms

Answer: 100 mV, 400 mV.




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Example Obtain the energy stored in each capacitor in Fig (a) under dc conditions

Solution:

Under dc conditions, we replace each capacitor with an open circuit, as shown in Fig. (b). The current through the series combination of the 2KΩ and 5KΩ resistors is obtained by current division as

Practice Problem Under dc conditions, find the energy stored in the capacitors in Fig.

Fundamentals of Electric Circuits - University of Anbar · 2020. 3. 14. · Fundamentals of EE II...

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